A Blast-Resistant Method Based on Wave Converters with Spring Oscillator for Underground Structures

Researches on blast-resistant measures for underground structures such as tunnels and underground shopping malls are of great importance for their significant role in economic and social development. In this paper, a new blast-resistantmethod based onwave converters with spring oscillator for underground structures was put forward, so as to convert the shock wave with high frequency and high peak pressure to the periodic stress wave with low frequency and low peak pressure. The conception and calculation process of this new method were introduced. The mechanical characteristics and motion evolution law of wave converters were deduced theoretically. Based on the theoretical deduction results and finite difference software FLAC3D, the dynamic responses of the new blast-resistant structure and the traditional one were both calculated. Results showed that, after the deployment of wave converters, the peak absolute values of the bending moment, shear force, and axial force of the structure decreased generally, which verified the good blast-resistant effect of the new blast-resistant method.


Introduction
Tunnels, subway stations, underground shopping malls, and so forth play an important role in the economic and social development, as well as the personal and property safety.In recent years, there have been a variety of researches on the blast-resistant measures for underground structures, mainly focused on the design and optimization on structures and materials.
Blast-resistant measures based on the structure optimization mainly contain the increase of stiffness or adoption of structures good for the reflection, diffraction, and scattering of the stress wave.Usually the arch structures and structures containing holes have better performance of wave dissipation than rectangular structures or structures without holes, so these kinds of structures have attracted many attentions of scholars [1][2][3].A good blast-resistant performance is also embodied in the plate-foam composite structure [4], boxshaped steel plate-reinforced concrete composite structure [5], carbon fiber reinforced composite structure [6,7], and prestressed structure [8].Via the explosion model tests, Yang et al. studied the antiexplosion effect of prestressed cable reinforced caverns.The results showed that prestressed structures are good for antiexplosion [9].Kobielak et al. experimentally investigated the influence of an attenuation barrier on soil stresses and pressures acting on a buried silo, caused by underground explosions at different distances.Test results verified the blast-resistant effectiveness caused by the barrier composed of the cylindrical tubes [10].The dynamic response of a buried silo caused by underground explosion was also studied experimentally via the measured pressures [11].
Blast-resistant measures based on the material optimization are mainly concentrated on the development of porous or lightweight materials of low stiffness, and the materials are developing gradually from the traditional inorganic porous materials or lightweight materials to the polymer materials and porous metal materials currently.Materials such as the rigid polyurethane foam, polypropylene fiber concrete, rubber concrete, foam concrete, foamed aluminium, and steel fiber reinforced concrete are good choices for the blastresistant materials.Yakushin et al. investigated the properties of low-density rigid polyurethane foams with hollow glass microspheres.The tension and compression properties in relation to the content of microspheres were determined in their work [12].Alhozaimy et al. found that the flexural toughness and impact resistance showed an increase in the presence of polypropylene fibers [13].The mechanical properties of concrete containing tire-rubber particles were studied by Khaloo et al. [14].They found that, unlike plain concrete, the failure state in rubberized concrete occurred gently and uniformly and did not cause any separation in the specimen.In fact, this property is good for dynamic protection.Hernández-Olivares and Barluenga also studied the fire performance of the rubber-filled concrete [15].Lim et al. studied the compressive, splitting tensile, and flexural strengths of lightweight foamed concrete [16].The results have shown that the foamed concrete is a good choice for dynamic protection.
In the blast-resistant methods of the traditional structure optimization, the construction process is usually complex, and sometimes the function of structures may even be affected.The shock wave mainly consists of high frequency components.In the blast-resistant methods of porous or lightweight materials, the materials are easy to get damaged unrecoverably and have large deformation under the blasting load because of the low elastic modulus and the existence of the holes.Thereby the overall stability of underground structure and surrounding rock may be affected by the large deformation.
In order to improve the traditional antiknock methods, a new blast-resistant method based on wave converters with spring oscillator for underground structures is put forward in this paper.The new method mainly consists of an array of wave converters and a distribution layer.Firstly, the conception of the new method is introduced, including the formation of the wave converter and distribution layer.Secondly, the calculation process of dynamic responses for underground structures adopting the new blast-resistant method is presented.Thirdly, the mechanical characteristics and motion evolution law of the wave converter are derived, including the static constitutive relation of the wave converter, dynamic response partitioning of the wave converter, differential equation of motion for the spring oscillator, displacement transfer coefficient of the wave converter, and the stress inversion of the wave converter.A case study is also conducted to verify the applicability and rationality of the new method by comparing with the traditional structure.

Conception of New Blast-Resistant Method
The new method mainly consists of an array of wave converters and a distribution layer, shown in Figure 1.The wave converter includes 2 shells containing a length adjusting rod and a spring oscillator comprised of springs and a mass block, shown in Figure 2. The initial length of the wave converter can be adjusted via the adjustment of the length adjusting rod, which can control the prestress of the compression springs.Via the adjustment of the initial converter length, the wave converter can also be easily installed under different reserve space between the rock and tunnel roof.The wave converter is compressible when the load on the top of the converter exceeds the spring prestress.The distribution layer is composed of the material with relatively low wave impedance, shown in Figure 1.Multiple reflections caused by the periodic stress wave can occur in the distribution layer, which results in the further energy dissipation.
The new blast-resistant method combines such mechanisms as the spring deformation, inertia, and periodic vibration of the mass block to provide the resistance against the dynamic load.The self-support capacity of the surrounding rock can also be fully utilized.Via the wave converter, the shock wave with high frequency and high peak pressure can be transformed to the periodic stress wave with low frequency and low peak pressure.Thereby the shock wave is dispersed and materials under the converter can be prevented from crushing.Under the blasting load, the deformation process of the wave converter can be divided into such 3 periods as rapid loading stage, rapid unloading stage, and slow unloading stage.
The above 3 stages are determined by the relative displacement Δ  () between the top and bottom of wave converter.Under the impact loads propagating in the rock, the typical curve of Δ  () with time for the wave converter is shown in Figure 3.As is shown in Figure 3, the rapid loading stage corresponds to the sharp increasing period, and the rapid unloading stage corresponds to the fast decreasing period,  while the slow unloading stage corresponds to the slow change period.

Calculation Process of Dynamic
Responses for Structures with the New Blast-Resistant Method (1) Establish numerical models and acquire the data needed for the calculation of the wave converter's displacements in the rapid loading stage and rapid unloading stage.Model 1 without wave converters for the finite element analysis is set up, shown in Figure 4.
In model 1, the distribution layer, underground structure, and rock are established.The initial pressure caused by the wave converter on the distribution layer surface (at the wave converter bottom) is applied as the lower spring force divided by the cross-sectional area of the wave converter.Then the explosion position and blasting load should be applied.After that, under the blasting load, the finite element analysis method is used to calculate the vertical displacementtime curves  0 () and  2 () of the rock surface at the wave converter top and the distribution layer surface at the wave converter bottom, respectively.The monitoring positions for displacements are suggested to adopt the tops and bottoms of 9 wave converters along the width direction of the structure, shown in Figure 4. Then model 2 of the ground without any construction such as the structure, distribution layer, and wave converters is built up, shown in Figure 5.
The numerical analysis based on model 2 under the same explosion condition as model 1 is conducted to obtain the vertical rock pressure-time curve  0 () for the corresponding monitoring positions of the rock surface in model 1, shown in Figure 5.
(2) Solve the differential equation of motion for the spring oscillator and obtain the law of motion in the rapid loading stage and rapid unloading stage.In model 1, the vertical displacement-time curve  0 () of the rock surface at the wave converter top multiplied by a displacement transfer coefficient   (0 <   < 1) is considered as the vertical displacement-time curve  1 () of the wave converter top in the rapid loading stage and rapid unloading stage of the true situation.The true situation refers to the real underground structure with an array of wave converters and a distribution layer.The vertical displacementtime curve  2 () of the distribution layer surface at the wave converter bottom in model 1 is considered directly as the vertical displacement-time curve of the wave converter bottom in the rapid loading stage and rapid unloading stage of the true situation.In the true situation, the difference between  1 () and  2 () is the relative displacement Δ  () between the top and bottom of the wave converter.The wave converter is in the rapid loading stage before the relative displacement reaches the maximum, while it is in the rapid unloading stage during the sharp decrease period after the relative displacement reaches the maximum.
Taking the displacements of the wave converter top and bottom as the boundary condition, the differential equation of motion for the spring oscillator can be solved to obtain the law of motion in the rapid loading stage and rapid unloading stage.Then the stress-time curve of the wave converter top and bottom in above 2 stages can be gotten.
(3) Solve the differential equation of motion for the spring oscillator and obtain the law of motion in the slow unloading stage.According to step (2), the length of the wave converter, the displacement (or position coordinate) and velocity of the oscillator at the end of the rapid unloading step can be gotten.
Then the differential equation of motion for the spring oscillator in the slow unloading stage can be solved.
In the slow unloading stage, the length of the wave converter can be deemed as a constant.In that case the calculation results of the dynamic response are a little dangerous which would be safer for the structure design.After that, the stress-time curve of the wave converter top and bottom in the slow unloading stage can be gotten.
(4) Calculate the dynamic response of the underground structure.Firstly, the initial pressure  in caused by the wave converter on the distribution layer surface in model 1 should be deleted.Then the stress-time curves of the wave converter top and bottom in three stages are applied on the rock surface and distribution layer surface, instead of the wave converters.After that, the blasting load is applied and the dynamic calculation is conducted to get the dynamic response of the structure.

Static Constitutive Relation of the Wave
Converter. 0 is the initial length of the wave converter, while  is the shell thickness.So the initial clear length of the wave converter can be written as  0 =  0 − 2, and  0 is shown in Figure 6.
In Figure 6, the dashed line is the position under the static equilibrium, and the solid line is the position at any time under the dynamic load. 1 is the stiffness coefficient of the upper spring, whose length in free state is  10 .Δ 10 is the initial amount of compression for upper spring. 2 is the stiffness coefficient of the lower spring, and  20 is its length in free state.Δ 20 is the initial amount of compression for lower spring. and  3 are the mass and height of the mass block, respectively.  () is the vertical rock pressure on the wave converter top in the true situation, while  0 () is the vertical rock pressure at the same location in model 2. When the wave converter is installed with the initial length  0 , under static equilibrium, the geometric equation is established, yielding As for the mass block, the balance equation is Combining ( 1) and ( 2), the solutions are 1 and  2 are the force of upper and lower springs, respectively. is the cross-sectional area of the wave converter. is the cross-sectional length.If the weight of the wave converter's shells is ignored,  1 can be expressed as 2 can be expressed as

Dynamic Response Partitioning of the Wave Converter.
The rapid loading stage, rapid unloading stage, and slow unloading stage correspond to the rapid compression stage, rapid recovery stage, and slow recovery stage, respectively.In the slow recovery stage, the length of the wave converter can be deemed as a constant.The demarcation point of stage 1 and stage 2 is that the relative displacement reaches the maximum.
The duration time of stage 2 can be determined by the relative displacement-time curve.When the rock masses above the structure are in a wide range of elasticity state, the rapid unloading stage can not be ignored, but if the rock masses are in a wide range of plastic state, the rapid unloading stage can be ignored.
The computing time of stage 3 is advisable for 1 or 2 vibration periods.Via a large amount of computations, it is concluded that the computing time of stage 3 can be taken as 1 vibration period if stage 2 can not be ignored; otherwise it can be taken as 2 vibration periods.

Differential Equation of Motion in Rapid Loading Stage
and Rapid Unloading Stage. is the total number of wave converters along the width direction of the structure, and  is the serial number of the wave converter.For wave converter ,  0 () is the vertical displacement of the rock surface at the wave converter top in model 1, and  2 () ( = 1, 2, . . ., ) is the vertical displacement of the wave converter bottom in rapid loading stage and rapid unloading stage of the true situation. 1 () is the vertical displacement of the wave converter top in rapid loading stage and rapid unloading stage of the true situation, which can be expressed as  1 () =    0 ().  is the displacement transfer coefficient of the wave converter.Δ  () is the relative displacement between the top and bottom of the wave converter, which can be written as Δ  () =  1 () −  2 ().At the end of the rapid unloading stage is the initial state of the slow unloading stage, which can be deemed as the fixed-length vibration, shown in Figure 7.In Figure 7, the dashed line is the position under the static equilibrium in the fixed-length vibration, and the solid line is the position at the end of the rapid unloading stage.
As is shown in Figure 6, V  (),   (), and   () are oscillator velocity, oscillator acceleration, and position coordinate at the time of , respectively, yielding The length of the upper spring is The amount of the spring compression is The length of the lower spring is The amount of the spring compression is The oscillator acceleration yields Then the differential equation of motion for the spring oscillator is The above equation is an ordinary differential equation of the second order, which can be solved by the Runge-Kutta method of the fourth order.This equation can be transformed to following forms: Assuming that   () =    (), the above equations can be written as following ordinary differential equations of the first order According to the Runge-Kutta method of the fourth order [17,18], its numerical calculation formula can be represented as where Given the displacement boundary conditions of the wave converter top and bottom ( 1 (),  2 ()), the position coordinate   () and velocity V  () of the oscillator at the time of  can be derived.Therefore, the amount of the compression Δ 1 (), Δ 2 () can be obtained.

Differential Equation of Motion in Slow Unloading Stage.
In the slow unloading stage, the vertical displacements of the wave converter top and bottom have few changes over time, so the length of the wave converter can be considered as a constant.  0 and   0 are the length and clear length of the wave converter at the end of the rapid unloading stage, respectively.As is shown in Figure 7, the geometry of the wave converter at the end of the rapid unloading stage is drawn in the solid line, while the dashed line shows the static equilibrium position of the mass block.The -axis is vertical downward and its coordinate origin is the static equilibrium position of the mass block in stage 3.In Figure 7, at the end of the rapid unloading stage, V 1 is the oscillator velocity, and  1 is the distance from the oscillator center to that of the static equilibrium position.
If the oscillator is in static equilibrium under the converter length of   0 ,   0 yields The balance equation for the mass block is where Δ 11 and Δ 21 are the amount of compression for upper and lower spring in static equilibrium under the converter length of   0 , respectively.Δ 11 and Δ 21 are ,   , and   are the active force, inertia force, and constraint counterforce, respectively.Based on the D' Alembert principle,   ,   , and   yield The active force consists of the gravity force , damping force   , and elastic restoring force   .Then (22) can be written as () is the position coordinate of the oscillator in  coordinate.The time at the end of the rapid unloading stage is assumed as   , and then the position coordinate and velocity of the oscillator can be expressed as   (  ) and V  (  ) in  coordinate.Via the coordinate transform, the position coordinate and velocity of the oscillator in  coordinate can be expressed as  1 and V 1 . 1 and V 1 are the initial conditions of (23), so the solution of ( 23) is where is the damping ratio.If  = 0, the calculation results of the dynamic response are a little dangerous which would be safer for the structure design, so  is considered as 0 in the analysis of stage 3. Considering that   () is solved, the amount of compression for springs at any time can be obtained.After that, the stress-time curve of the wave converter top and bottom in the slow unloading stage can be gotten.

Displacement Transfer Coefficient of the Wave Converter.
In the rapid loading stage and rapid unloading stage, based on the numerical calculations in model 1 and model 2, the following equation can be derived: where  is the area ratio of the wave converter's crosssectional area, and  is equal to all wave converters' crosssectional areas divided by the area of rock surface above the structure.According to (4) and (7), when   () ≥  1 Δ 10 , the constitutive relation for the wave converter can be modified as Combining ( 26) and ( 27),   yields

Stress Inversion of the Wave Converter.
Based on the solutions on differential equations of motion in 3 stages, if the gravity force of the wave converter shell is ignored, the stresstime curves of the wave converter top and bottom in 3 stages can be gotten.
In stage 1 and stage 2, the function of the stress-time curve of the wave converter top is  The function of the stress-time curve of the wave converter bottom is In stage 3, the functions of the stress-time curves of the wave converter top and bottom are, respectively, Then, in model 1, the initial pressure  in caused by the wave converter on the distribution layer surface is deleted, and the stress-time curves of the wave converter top and bottom in three stages are applied on the rock surface and distribution layer surface, respectively.After that, the blasting load is applied and the dynamic calculation is conducted to get the dynamic response of the structure (shown in calculation process (4)).

Case Study
5.1.Case Set-Up.Based on the software of FLAC 3D , 2 kinds of cases are set up to conduct the dynamic analysis.Case 1 is the traditional underground structure without the new blastresistant method, while case 2 is the underground structure with the mentioned new blast-resistant method.Compared with case 2, the difference in case 1 is that the wave converter and distribution layer are not set.Figure 8 shows the structure size and monitoring sections for internal forces in case 2, and the monitoring sections for internal forces are the same as case 1.The size of the structure is designed according to literature [19], and the buried depth is 10 m.Mechanical parameters for the rock and structure are listed in Tables 1 and 2. The distribution layer is made up of the foam concrete which has the density of 799 kg/m 3 and thickness of 0.05 m.The mechanical parameters of the foam concrete are shown in Table 3 [20].In case 2, the tops and bottoms of wave converters 1∼5 are chosen as monitoring positions for displacements and stresses, shown in Figure 8, and the interpolation is used to get the stresses of other wave converters.The wave converter is a cubic structure with the side length of 0.2 m.  0 = 0.2 m,  = 0.05 m,  = 100%,  1 =  2 = 100 kN/m,  10 =  20 = 0.05 m.  3 = 0.14 m,  = 38.9kg.
The width, height, and thickness of the numerical models are 35.2m, 28.85 m, and 1 m, respectively.The blasting load is assumed as a triangle wave acting on the ground surface (in Figure 9), and the loading scope is from −3 m to 3 m on -axis.The lifting duration and drop duration of the blast pressure-time curve are set as 1 ms and 6 ms according to the literature [21], respectively.The peak of the shock wave   pressure   is 0.6 GPa.Model 1 built up based on FLAC 3D for numerical calculation is shown in Figure 10.
According to the symmetry, the monitoring positions for displacements are suggested to adopt the tops and bottoms of 5 wave converters along the width direction of the structure, shown in Figure 8.In order to obtain the internal forces such as the bending moment, axial force, and shear force of the structure, a total of 15 monitoring sections are set up, shown in Figure 8.  4-6.

Analysis of the Calculation
In Tables 4-6, the bending moment resulting in the tensile stress in the inner element of the structure is positive.The positive shear force is by counterclockwise, while the axial force to tension is positive.It can be concluded that the peak absolute value of the bending moment in case 2 is generally lower than that of case 1, and the maximum drop in the roof, side wall, and floor is 57.3%, 69.7%, and 52.7%, respectively.
The peak absolute value of the shear force for monitoring sections in case 2 is also generally lower than that of case 1.The maximum drop in the roof, side wall, and floor is 86.2%, 75.6%, and 33.1%, respectively.The peak absolute value of the axial force for monitoring sections in case 2 is remarkably lower than that of case 1.The maximum drop in the roof, side wall, and floor is 44.7%, 74.7%, and 37.2%, respectively.For the roof, the decrease of the axial tensile force near the midspan is obvious, and the maximum drop occurs to the  span center.The decrease of the axial tensile stress presents that the tensile failure in the roof can be alleviated via the wave converters.
The curves of the vertical normal stresses for inner and outer elements in the span center with time are shown in Figure 18.The positive value stands for the vertical tensile stress, while the negative value stands for the vertical compressive stress.After the adoption of wave converters, the peak vertical tensile stress for inner element in the span center drops from 0.055 MPa to 0.019 MPa, which means that the possibility of spalling damage for roof is reduced.The peak vertical compressive stress for outer element in the span center drops from 1.73 MPa to 0.153 MPa, which means that the impact load acting on the roof is reduced.
The peak horizontal tensile stresses of monitoring sections for roof in 2 cases are shown in Figure 19.For case 2  with wave converters, the peak horizontal tensile stresses of inner elements for roof are generally lower than that of case 1, with a maximum decrease of 60%.Though the peak horizontal tensile stresses of outer elements for roof become higher than that of case 1, the peak horizontal tensile stresses after increasing are not very large on the whole.Via increasing the spring stiffness, the increase of the peak horizontal tensile stresses of outer elements for roof can be adjusted.In the design process of wave converters, the vibration period of the spring oscillator should not be next to the vibration period of the underground structure.

Conclusions
In this paper, a new blast-resistant method based on wave converters with spring oscillator for underground structures is put forward.The conception and calculation process of this

Figure 1 :Figure 2 :
Figure 1: Schematic of the new blast-resistant method.

Figure 6 :
Figure 6: Deformation process of the wave converter.

Shock and Vibration 5 So
the static constitutive relation of the wave converter yields If   () <  1 Δ 10 ,

Figure 7 :
Figure 7: Initial state of the fixed-length vibration.

Figure 8 :
Figure 8: Structure size and monitoring sections for internal forces in case 1.

Figure 9 :
Figure 9: Curve of the blasting load.

Figure 10 :
Figure 10: Model 1 built up based on FLAC 3D for numerical calculation.

Figure 11 :
Figure 11: Curves of the displacement transfer coefficient with time.

Table 1 :
Mechanical parameters for the rock.

Table 2 :
Mechanical parameters for the structure.

Table 3 :
Mechanical parameters for foam concrete.

Table 4 :
Internal forces of the structure for monitoring sections 1-5.

Table 5 :
Internal forces of the structure for monitoring sections 6-10.